Electron heating of over-dense plasma with dual-frequency electron cyclotron waves in fully non-inductive plasma ramp-up on the QUEST spherical tokamak

The equilibrium of the 25 kA over-dense plasma obtained through SDJs in the dual-f  injections was analyzed using EFIT code. Figure 8 shows the equilibrium magnetic surface structures of the 25 kA over-dense plasma before and after the second SDJ (at t  =  1.85 s and 2.25 s). Large Shafranov shifts were seen at both timings; $R_{{\rm ax}}$ and $R_{{\rm out}}$ decreased after the second SDJ, in agreement with the filament current analysis and the TS measurement result. Dense flux surface structures appeared between $R_{{\rm ax}}$ and $R_{{\rm out}}$. The self-organized plasma shaping with small elongation $\kappa$ was quite different from the shaping of the plasma ramped up with solely the 28 GHz wave [12]. The poloidal beta $\beta^{*}_{{\rm p}}$ in the plasma using the dual-f  waves (∼1.5) was larger than for the 28 GHz plasma (∼0.8). The self-organized plasma shaping has been discussed in regard to high $\beta_{{\rm p}}$ plasmas [26].

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The plasma or electron pressure required for the equilibrium was larger than 400 Pa near the axis and, being a function of nested flux surfaces, its profile was bell-shaped. In contrast, the bulk pressure measured by the TS diagnostics was  ∼10 Pa and its profile was hollow-like. In the EFIT analysis, the pressure and current profiles of low-order polynomials were used with the external magnetics for constraints; the distribution profiles therefore cannot be discussed in detail.

Electron heating effects with the 8.2 GHz wave on energetic and bulk electrons after the second SDJ are discussed next. Figure 9 shows the radial profiles of electron cyclotron frequency $f_{{\rm ce}}(R)$, plasma frequency $f_{{\rm p}}(R)$, and upper hybrid resonance (UHR) frequency $f_{{\rm UHR}}(R)$ on the 25 kA over-dense plasma at t  =  2.55 s. An 8.2 GHz wave reached the O-mode cutoff layer $R^{\ {\rm O}}_{{\rm cut}} \sim 0.93$ m where f   =  $f_{{\rm p}}$, and the O-mode wave converted into the X-mode. The wave propagated to the upper hybrid resonance layer of $R_{{\rm UHR}}\sim 0.99$ m after the O–X mode conversion, and then the X-mode converted into EBW at $R_{{\rm UHR}}$. The electrostatic EBW can propagate to the inner (high-density) side beyond $R \sim R^{\ {\rm O}}_{{\rm cut}}$. The resonance condition at R  =  0.93 m, where the significant $P^{\ {\rm TS}}_{{\rm e}}$ increase was observed, was considered in regard to the 8.2 GHz heating effect for bulk and/or mildly energetic electrons. Although the position of R  =  0.93 m is significantly far from $R_{\ {\rm 1st}}^{\ {\rm 8.2}}$, the bulk and mildly energetic electrons are expected to be in resonance with the electrostatic wave of relatively large parallel refractive index $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} | N_{\para}~|( > 1)$ to the magnetic field, as discussed below. The position of R  =  0.93 m was roughly half the minor radius, a/2.

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The nth harmonic relativistic Doppler-shifted ECR condition with parallel refractive index $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ is introduced. The nth harmonic ECR condition is expressed as $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \gamma[1-N_{\para}\ (v_{\para} /c)] = n f_{{\rm ce}} / f$ with relativistic factor $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \gamma=(1+\overline{P}_{\perp}^{2} +\overline{P}_{\para}^{2}){}^{1/2}$, where $\overline{P}_{\perp}=m\gamma v_{\perp} / (mc)$ and $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para}=m\gamma v_{\para} / (mc)$ are the normalized electron momenta in the perpendicular and parallel directions to the magnetic field, respectively. Here, $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} v_{\para}$ and f  are the parallel electron velocity and operating frequency, respectively. Furthermore, m is the electron rest mass, and c is the speed of light. The ECR condition can be rewritten as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \displaystyle (1-N_{\para}^{2})\overline{P}_{\para}^{2} -2 N_{\para} \frac{n f_{{\rm ce}}}{f} \overline{P}_{\para}+1+\overline{P}_{\perp}^{2} -\Bigl( \frac{n f_{{\rm ce}}}{f} \Bigr)^{2}=0. \nonumber \end{align} \tag{ 2 }$$

As is well known, this equation can be reduced to an elliptical equation of [$\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para}$, $\overline{P}_{\perp}$] when $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|< 1$ [12, 27], whereas it is reduced to parabolic and hyperbolic equations when $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|=1$ and $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|> 1$, respectively [28].

The parallel refractive index $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ is expressed as $[ N_{{\rm t}} B_{{\rm t}}+ N_{{\rm p}} B_{{\rm p}} ] / B$, where $N_{{\rm t}}$ and $B_{{\rm t}}$, and $N_{{\rm p}}$ and $B_{{\rm p}}$ are the refractive indexes and magnetic field components in the toroidal and poloidal directions, respectively; and B is the magnetic field strength. The toroidal component of $N_{{\rm t}} B_{{\rm t}} / B$ is normally dominant in the $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ evolution; however, the poloidal component of $N_{{\rm p}} B_{{\rm p}} / B$ can become significant in electrostatic waves. The perpendicular refractive index $N_{\perp}$ to the magnetic field should be less than unity for electromagnetic waves, whereas it can become large beyond unity for electrostatic waves. For instance, $N_{\perp}$ may be an order of $\sim c /(\rho_{{\rm e}} f)$ for EBW, where $\rho_{{\rm e}}$ is the Larmor radius of the electron. $N_{\perp}$ is linked to $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ through total magnitude of the refractive index N; specifically,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \displaystyle N_{\perp}^{2}=N^{2}-N_{\para}^{2 }=N_{{\rm r}}^{2}+N_{{\rm p}}^{2}+N_{{\rm t}}^{2} - N_{\para}^{2}, \nonumber \end{align} \tag{ 3 }$$

where $N_{{\rm r}}$ is a radial component of the refractive index in the toroidal geometry. $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ would become large beyond unity as well as N and $N_{\perp}$.

Figure 10 shows, for the fundamental ECR conditions, various $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para} - \overline{P}_{\perp}$ plots for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s at R  =  0.93 m where $f_{{\rm ce}} / f=0.59$. The minimum $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N^{{\rm min}}_{\para}|$ satisfying the resonance condition is expressed as $\sqrt{1 - (\,f_{{\rm ce}} / f){}^{2} }$, and then $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} | N^{{\rm min}}_{\para}|$  =  0.81 at R  =  0.93 m. In the figure, small ellipses, each with a relatively large offset of the origin $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \newcommand{\e}{{\rm e}} \overline{P}_{\para\ {\rm O}}\ [ \equiv N_{\para}\ (\,f_{{\rm ce}} / f) / ( 1 - N_{\para}^{2})]$ of 1.8, appear for $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}=\pm 0.85$. The major ($\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para})$ and minor ($\overline{P}_{\perp})$ radii, $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} a_{\para}$ and $a_{\perp}$, of the ellipse are defined as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \displaystyle a_{\para} = \sqrt{\Bigl( \frac{f_{{\rm ce}}}{f}\Bigr)^{2} - (1-N^{2}_{\para})\ }/\ \ \biggl[ {1-N^{2}_{\para}} \biggr], \nonumber \end{align} \tag{ 4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \displaystyle a_{\perp} = \sqrt{\Bigl( \frac{f_{{\rm ce}}}{f}\Bigr)^{2} \frac{1}{1-N^{2}_{\para}}-1}. \nonumber \end{align} \tag{ 5 }$$

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With increasing $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ $( < 1)$, the radii and origin offsets of the ellipse become large. The parabolic curves for $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  $\pm1$ with x-axis intercepts $\newcommand{\e}{{\rm e}} (\equiv \pm [ 1 - (\,f_{{\rm ce}} / f){}^{2} ] / [ 2 (\,f_{{\rm ce}} / f) ])$ of $\pm$ 0.55 are also shown in the figure. In addition, hyperbolic curves with x-axis intercepts of small and large magnitudes $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \newcommand{\e}{{\rm e}} (\equiv | \overline{P}_{\para\ {\rm O}} \pm a_{\para}~|)$ are plotted for $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  $\pm 1.2$. As the Doppler-shifted EC resonance effect is taken into account through the product $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}\ v_{\para}$, electrons with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm P_{\para}$ are in resonance with waves for $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm N_{\para}$ under ray-scrambling situations, whereas there is an asymmetry of $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm v_{\para}$ at a specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$.

If an $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ sign is specified with the focusing and steering beam, the resonance $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} P_{\para}$ range becomes asymmetric on its $\pm$ region, in particular, to the electrostatic wave resonance as suggested from the hyperbolic resonance structure with small and large magnitudes of x-axis intercepts. The asymmetry of the hyperbolic resonance is much stronger than that of the elliptical resonance. Therefore, the plasma current can be driven more effectively by the EBWHCD with a hyperbolic resonance than by the ECHCD with an elliptical resonance, if the incident wave is properly absorbed at specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. The $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ evolution of the electrostatic wave notably depends on the propagating poloidal position and its direction, because the poloidal component $N_{{\rm p}} B_{{\rm p}} / B$ becomes significant in the evolution. If the launching aperture and the propagating poloidal position expand over the upper and lower sides for the mid-plane, the rays are not easily absorbed at specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s after mode conversion to the EBW [18, 28].

During the non-inductive plasma-current start-up, a Pfirsch–Schlüter current may be created as a seed current on the open field lines if the vertical motion from toroidal drift $v_{{\rm d,t}}$ is cancelled by the vertical component of the parallel velocity $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \sim v_{\para} (B_{{\rm v}}/B_{{\rm t}})$ determined by the magnetic field pitch angle [29]. Therefore, there is an asymmetric property of $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm P_{\para}$ imposed on the electron confinement in the current start-up phase at the open field lines. Moreover, a different radial shift or deviation of the passing electrons with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm P_{\para}$ from the magnetic surface contributes to the plasma-current ramp-up in the inner limiter configuration after the formation of closed flux surfaces. The ramped plasma current is enhanced by the ECW even in the ray-scrambling situation with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \pm N_{\para}$, as $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} P_{\para}$ has an asymmetric distribution in the $\pm$ directions in the start-up phase. Banana orbits are possible even in open field lines with a magnetic mirror with $M (> 1)$. A relatively high $M (> 1)$ favors the confinement of energetic electrons generated by the ECW. The confined trapped electrons also contribute to current generation through precession in the toroidal direction. The electron heating effect and mechanism by the ECW and EBW should be properly assessed for such non-inductively EC ramped-up plasmas.

In regard to bulk electron heating, figure 11(a) shows a semicircle with constant energy of 50 eV as well as some resonance hyperbolic curves in the [$\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para}$, $\overline{P}_{\perp}$] plane. The intersection point between the resonance curve and the constant-energy semicircle indicates the resonance condition of the electron for a specific energy. Because the semicircle radius of $\sqrt{\gamma^{2}-1}$ was so small for the 50 eV electron, there was no intersection point with the resonance ellipse when $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~| < 1$. The hyperbolas with larger $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|\ ( \gtrsim 30)$ may have intersection points with the constant-energy semicircle. These 50 eV electrons are in resonance with the electrostatic waves for $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|\ ( \gtrsim 30)$. Although $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|$ may be larger than unity, the required $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|$ is an order of magnitude greater than that expected from previous ray-tracing analyses [18, 28] with various incident $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. It is unlikely the electrons were directly heated by the 8.2 GHz electrostatic wave in the over-dense region, even if the tail components of the Maxwell distribution were in resonance with the wave for somewhat smaller $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} |N_{\para}~|$.

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The AXUV diagnostic, which is sensitive in the SX energy range (>0.3 keV), showed extremely enhanced signals in the over-dense plasma. The EBW heating effect on mildly energetic electrons was considered here. The constant-energy semicircles for 5 and 10 keV electrons are plotted in figure 11(b), together with some resonance hyperbolas for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. Such mildly energetic electrons may be in resonance with the electrostatic waves for moderate $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} | N_{\para}~|$ of 2–3, as shown in the figure. The possibility that mildly energetic electrons were directly heated by the 8.2 GHz electrostatic wave with reasonable $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ is indicated, taking the resonance condition into account. Moreover, whether such electrostatic waves may be excited should be considered. The ray-tracing approach was not available for consideration of the wave excitation under the ray-scrambling situation with multiple wall reflections.

Concerning wave excitations, dispersion with the bulk and energetic electron components was considered. The dispersion D is expressed in the electrostatic approximation [30] in the form,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle D=& 1 +\Biggl[ \frac{2X}{N^{2} \mu^{2}} \Bigl[ 1+ \xi_{0}\ \sum^{+\infty}_{n=-\infty} {\sf Z}(\xi_{n}) {\rm e}^{-\lambda} I_{n} (\lambda)\ \Bigr]\ \Biggr]_{{\rm b}} \nonumber \\ & \ + \Biggl[ \frac{2X}{N^{2} \mu^{2}} \Bigl[ 1+\xi_{0}\ \sum^{+\infty}_{n=-\infty} {\sf Z} (\xi_{n}) {\rm e}^{-\lambda} I_{n} (\lambda)\ \Bigr]\ \Biggr]_{{\rm h}}, \nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \displaystyle \mu=& \frac{v_{{\rm th}}} {c}=\frac{\sqrt{2 k_{{\rm B}}T_{{\rm e}}/m}}{c}, \ \ \xi_{n}=\frac{1-nY}{N_{\para}\mu}, \nonumber \\ X=& \frac{f_{{\rm p}}^{2}}{f^{2}},\ \ Y=\frac{f_{{\rm ce}}}{f},\ \ \lambda =\frac{1}{2} \bigl( \frac{\mu N_{\perp}}{Y} \bigr)^{2}, \nonumber \end{align} \tag{ 7 }$$

where $k_{{\rm B}}$ is the Boltzmann constant, and $T_{{\rm e}}$ is the electron temperature. The plasma dispersion function ${\sf Z}$ is used in the non-relativistic limit, and I_n is the nth-order modified Bessel function. Here, the subscripts (b) and (h) signify that the associated quantity refers to the bulk and energetic electron components, respectively. The bulk electron temperature $T_{{\rm eb}}$ is set to 50 eV, and the proper $N_{\perp}$ giving D  =  0 is determined for a specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ and energetic electron temperature $T_{{\rm eh}}$. Furthermore, as the characteristic wave-damping length is 2 $|{\rm Im}D|$, the imaginary part of D is evaluated to take into account heating effects on the energetic electrons considered.

Figure 12 shows the evaluated dependences of $N_{\perp}$ and $|{\rm Im}D|$ on the energetic electron temperature $T_{{\rm eh}}$ for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s at R  =  0.93 m and 0.80 m. The $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ range of (1,3] was considered for electrostatic waves with hyperbolic resonance conditions. The energetic electron density $n_{{\rm eh}}$ was taken to be 3% of the bulk electron density $n_{{\rm eb}}$. Because $\mu_{{\rm h}}$  =  0.1 at $T_{{\rm eh}}\sim 2.6$ keV, a larger range than this value is marginal in the non-relativistic limit. Given a proper $N_{\perp}$ for a specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$, when the dispersion D becomes zero, a wave with a specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ and $N_{\perp}$ may be excited in the plasma. As the $N_{\perp}$ is mainly determined by the bulk electron component $[\ \cdots \ ]_{{\rm b}}$, $N_{\perp}$ should be almost constant despite various $T_{{\rm eh}}$ values. The imaginary part $|{\rm Im}D|$ is expressed with the imaginary part of $|{\sf Z}_{{\rm h}}|$, $|{\rm Im}{\sf Z}_{{\rm h}}|$, depending on $\newcommand{\e}{{\rm e}} \exp (-\xi_{n,\ {\rm h}}^{2})$. The magnitude of the imaginary part increases as $T_{{\rm eh}}$ increases, because the exponent magnitude of $\newcommand{\e}{{\rm e}} \exp (-\xi_{n,\ {\rm h}}^{2})$ decreases with increasing $T_{{\rm eh}}$. It then starts to decrease as $\newcommand{\e}{{\rm e}} \exp ( - \lambda_{{\rm h}})$ with larger $\lambda_{{\rm h}}$ when $T_{{\rm eh}}$ increases further. The exponent magnitude $|\xi_{n, {{\rm h}}}|$ is normally large because of the smallness of $\mu_{{\rm h}}$; however, when $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}\ \mu_{{\rm h}}\sim (1-nY)$, $\xi_{n, {{\rm h}}}\sim1$. The condition $\xi_{n}=1/\sqrt{2}$ (at $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} v_{\para} = v_{\perp}$) corresponds to the nth Doppler-shifted resonance condition in the non-relativistic limit. In general, the $\mu_{{\rm h}}$ or $T_{{\rm eh}}$ range with significant wave absorption diminishes when taking larger $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$, as expected from the hyperbolic resonance condition, because the factor $\xi_{0, {{\rm h}}}$ and the argument $\xi_{n, {{\rm h}}}^{2}$ of ${\sf Z}_{{\rm h}}$ are expressed with the product $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}\ \mu_{{\rm h}}$.

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First, the results of the dispersion analysis at R  =  0.93 m are discussed in regard to the bulk electron heating effect observed in the TS measurements. A weak dependence of $N_{\perp}\sim 22$–23 on $T_{{\rm eh}}$ was reasonably obtained for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. The $T_{{\rm eh}}$ range with significant $|{\rm Im} D|$ is reduced for larger $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$. $|{\rm Im}D|$ may grow to a large value for larger $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$, because $\newcommand{\e}{{\rm e}} \exp (-\lambda_{{\rm h}})$ does not dominate at lower $T_{{\rm eh}}$. Although the hyperbola with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} | N_{\para}~|\gtrsim$ 2 crosses the 10 keV constant-energy circle, the wave with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  2 may start to be absorbed even in the energy range less than 0.5 keV, as suggested from the $|{\rm Im}D|$ analysis. The waves may be absorbed by the tail electrons of the relatively lower than expected $T_{{\rm eh}}$ from resonance condition analysis. Mildly energetic electrons of around 1 keV may be heated by the electrostatic EBW with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para} \sim 2$ at R  =  0.93 m. Thus, the bright AXUV signal after the second SDJ can be explained by the EBW heating effect on the mildly energetic electrons of  ∼1 keV in the over-dense region.

The bulk electron heating suggested from the TS measurements was considered from the point of view of the relaxation processes from mildly energetic (1 keV) to bulk (50 eV) electrons. The heat-exchange time $\tau_{{\rm bhe}}$ is about 16 ms, and its relaxation power density $3\ n_{{\rm eb}} [ T_{{\rm eh}} - T_{{\rm eb}} ] / ( 2 \tau_{{\rm bhe}})$ may be significant (∼12 kWm⁻³) in heating the bulk electrons. The increments of $T^{\ {\rm TS}}_{{\rm e}}$ and $P^{\ {\rm TS}}_{{\rm e}}$ in the over-dense region may be due to heat exchange from the mildly energetic electrons heated by the electrostatic EBW. There was concern over the linkage of electron heating between the inner (high-field) and outer (low-field) sides of the same flux surface. However, concerning the heating effect, as $R_{{\rm ax}}$ shifts inward after the second SDJ, the outer position of R  =  0.93 m is probably not on the same flux surface as the inner position with high $T^{\ {\rm TS}}_{{\rm e}}$ near $R^{\ 28}_{\ {\rm 2nd}}$. The observed bulk electron heating may not be explained by the 28 GHz ECHCD effect and its related effects.

A weak bulk electron heating effect was observed at R  =  0.80 m. Hence, the results of dispersion analysis at R  =  0.80 m are assessed in comparison with those at R  =  0.93 m. The parameters X and Y with $f_{{\rm p}}$ and $f_{{\rm ce}}$ at R  =  0.80 m were used in this analysis. The perpendicular refractive indexes $N_{\perp}$ s $\sim 51$–52 were evaluated with weak dependence on $T_{{\rm eh}}$ for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. Because $N_{\perp}$ or $\lambda_{{\rm h}}$ becomes large at R  =  0.80 m, $| {\rm Im}D |$ is significantly diminished in terms of $\newcommand{\e}{{\rm e}} \exp ( - \lambda_{{\rm h}})$. The evaluated $| {\rm Im}D |$ of R  =  0.80 m is an order of magnitude less than that of R  =  0.93 m (figure 12). Although the wave absorption has to be evaluated from the wave-damping characteristic length and its path length, in actuality, the small $|{\rm Im}D|$ may be key in explaining the difference between the heating effects at R  =  0.93 m and 0.80 m.

Backscattering instability with parametric decay may prevent the wave from mode-converting into the EBW at the UHR [31]. However, the power-density threshold on the instability becomes high when electron temperature at the UHR $T_{{\rm e}}^{\ {\rm UHR}}$ is high, because the instability is suppressed by Landau damping for lower hybrid waves generated at the parametric decay. As we have not attempted to measure the parametric decay instabilities (PDIs), whether PDIs have occurred or not is uncertain. In the present dual-f  experiment, $T_{{\rm e}}^{\ {\rm UHR}}$ is relatively high, because the UHR position $R_{{\rm UHR}} \sim 0.99$ m is near the outer position $R \sim 1.0$ m on the magnetic flux surface of $R_{\ {\rm 2nd}}^{\ {\rm 28}}$ at the inboard side. It is likely that the significantly low 8.2 GHz power density under the ray-scrambling situation due to multiple wall reflections did not exceed the power-density threshold with relatively high $T_{{\rm e}}^{\ {\rm UHR}}$.

The 8.2 GHz heating effects on bulk and mildly energetic electrons were assessed at R  =  0.80 m and 0.93 m. The heating on highly energetic electrons of more than 60 keV was not considered because no significant absorption effects were evaluated from the dispersion analysis. However, the high $T_{{\rm rad}}^{\ {\ \rm HX}} \sim 500$ keV was actually observed in the FWD tangential viewing of $R_{{\rm tan}}$  =  0.71 m with the over-dense region, but without the off-axis region near $R^{\ 28}_{\ {\rm 2nd}}$ of 0.32 m. In contrast, $T^{\ {\ \rm HX}}_{{\rm rad}}$ was  ∼230 keV in the FWD viewing of $R_{{\rm tan}}$  =  0.34 m with the off-axis region. It is likely that there are abundant highly energetic electrons in the over-dense core region, as suggested from the EFIT equilibrium analysis.

The resonance conditions on the highly energetic electrons $\sim 500$ keV were considered for the 28 GHz wave. For the notably high energetic electrons, higher harmonic heating effects must be properly considered. The third (3rd) 3$f^{\ 28}_{{\rm ce}}$ and fourth (4th) 4$f^{\ 28}_{{\rm ce}}$ resonance positions are at $R^{\ 28}_{\ {\rm 3rd}}$  =  0.48 m and $R^{\ 28}_{\ {\rm 4th}}$  =  0.64 m, respectively. Figure 13 shows the dependences of the second, third, and fourth resonance momentum pitches on R, [$\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} \overline{P}_{\para}$ / $\overline{P}_{\perp}$] (R), of the 500 keV electrons for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. The normalized resonance momentum pitches were derived from intersection points of the resonance ellipses and a 500 keV constant-energy semicircle. The positive pitches were evaluated for up-shifted relativistic Doppler-shift resonances with positive $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ in this configuration. The waves that could access the core region near $R_{{\rm ax}} \sim 0.8$ m were considered. A couple of the parallel indexes ($\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  0.2, 0.5 and 0.8) were taken at R  =  0.80 m, and the $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ evolution maintaining $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} RN_{\para}~=~{\rm const}.$ was considered along the propagation. A wave for which $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ approaches unity along the propagation cannot access the inner high-field side further in the low-field side injection. For instance, the wave with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  0.8 at R  =  0.80 m cannot propagate beyond R  =  0.64 m. The wave trajectory becomes tangential ($\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  1) at R  =  0.64 m and then goes out to the low-field side. In the high-field side access after reflection at the inboard limiter, the wave propagates easily to the low-field side. The $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ of unity at $R^{\ {\rm lim}}_{{\rm in}}$  =  0.22 m becomes 0.275 in propagating to R  =  0.80 m while maintaining $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} RN_{\para}$  =  ${\rm const}$., all waves after the reflection at the inboard limiter can reach R  =  0.80 m and $R_{{\rm out}}$.

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The plotted resonance momentum pitch curves must cut across a zero pitch at a specific but same R for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s, depending on the harmonic number. For instance, the fourth resonance pitch curves for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s cut cross the zero pitch around $R^{\ 28}_{\ {\rm 2nd}}$. A broad extent of the Doppler-shifted regions around the relativistic shifted (zero-pitch) position is shown for larger $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ cases mainly in the up-shifted resonance side, because the waves cannot propagate to the high-field or the down-shifted side beyond R with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$  =  1 in propagation from the low-field side. The up-shifted resonance contributes effectively to the current drive if there is no down-shifted resonance in propagation from the low-field side, because the down-shifted absorption normally cancels out the current driven by the up-shifted absorption. In general, even if the incident beam is properly controlled to obtain significant single-path absorption, the down-shifted absorption after the up-shifted resonance reduces the current drive efficiency if the up-shifted absorption is not perfect.

Resonance overlapping between different cyclotron harmonics appears in the core region. The resonance momentum pitches depend on the harmonic number. Although the absorption at the lower harmonic resonance is strong, synergetic heating is expected because of overlapping resonances. Such heating along with higher harmonic resonances is one of the candidates for the heating and maintenance mechanism of the highly energetic electrons in the core region. The growth process of the highly energetic electrons may be considered with the simultaneous heating effect of the dual-f  wave in the low-density phase. The energetic electrons  ∼80 keV may be generated with the 8.2 GHz wave in the core region near $R^{\ 8.2}_{\ {\rm 1st}}$ at the low-density state, as expected from the previous 8.2 GHz EC plasma ramp-up experiments. The incident 28 GHz wave must be significantly absorbed by the energetic electrons at the higher (third and fourth) harmonic ECRs. Therefore, the generated energetic electrons may be accelerated more in the low-density phase with the dual-f  waves. This heating effect with the dual-f  waves may be verified if the polarized incident beams are properly focused with specific $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s for the dual-f  waves, a topic that will be addressed in future work.

The TS density $n^{\ {\rm TS}}_{{\rm e}}$ already exceeded $n^{\ {\rm 8.2}}_{{\rm e, cutof\,\!f}}$ at R  =  0.64 m even at t  =  1.85 s before the second SDJ, and the core density rose rapidly during the SDJ. Concerning the SDJ process, the plasma produced effectively in the core region near $R^{{\rm\ 8.2}}_{\ {\rm 1st}}$ was notably a higher-density plasma, probably formed through wave–plasma and/or some interactions of the bulk electrons. The electron heating effect on the effective density rise was considered through the dispersion analysis at R  =  0.64 m. The bulk electron temperature $T_{{\rm eb}}$ was fixed at 20 eV in this analysis. The parameters Y was evaluated with $f_{{\rm ce}}$ at R  =  0.64 m, and two $X_{{\rm b}}$ parameters were used for the electron density of 1 $\times$ 10¹⁸ m⁻³ and 2 $\times$ 10¹⁸ m⁻³ before and after the second SDJ (at t  =  1.85 s and 2.55 s), respectively. Figure 14 shows the dependences of $N_{\perp}$ and $|{\rm Im}D|$ on the energetic electron temperature $T_{{\rm eh}}$ in the high- and low-density cases before and after the second SDJ at R  =  0.64 m for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. The bulk terms of $[\ \cdots \ ]_{{\rm b}}$ mainly determined $N_{\perp}$ with $X_{{\rm b}}$ s and Y at R  =  0.64 m. Relatively large $N_{\perp}$s (∼167 and 209 in the low- and high-density cases) were obtained with weak dependences on $T_{{\rm eh}}$ for several $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para}$ s. As $(1- n Y)$ is smaller at R  =  0.64 m near $R^{\ {\rm 8.2}}_{\ {\rm 1st}}$, the $T_{{\rm eh}}$ range with significant $| {\rm Im}D |$ diminishes. $N_{\perp}$ is significantly large, and therefore, $| {\rm Im}D |$ at R  =  0.64 m becomes smaller than that at R  =  0.80 m in terms of $\newcommand{\e}{{\rm e}} \exp (-\lambda_{{\rm h}})$. However, $|{\rm Im}D|$ for the high-density plasma was the same as that in the low-density plasma even with larger $N_{\perp}$. A larger $N_{\perp}$ satisfying D  =  0 in the high-density case was evaluated to compensate twice the $X_{{\rm b}}$ of the low-density case. When $X_{{\rm h}}$ increases linearly with increasing $X_{{\rm b}}$, for both low- and high-density plasmas, the factors in the independent ${\sf Z}$ on $N_{\perp}$ were also identical in the energetic electron component, thereby explaining the same absorption effects for both cases. A modeling of the increment in $X_{{\rm h}}$ together with $X_{{\rm b}}$ was supported from the extremely enhanced AXUV signals after the second SDJ. The heat-exchange time $\tau_{{\rm hbe}}$ decreased by half if $X_{{\rm h}}$ increased twofold with increasing $X_{{\rm b}}$. The heat-exchange power then quadrupled with the doubling of $X_{{\rm b}}$ and $X_{{\rm h}}$ in the high-density case, suggesting effective plasma production and maintenance with non-decreasing $|{\rm Im}D|$ even in the high-density plasma. The core $P^{\ {\rm TS}}_{{\rm e}}$ after the second SDJ increased by more than twice as much as before the second SDJ. Although $|{\rm Im}D|$ becomes smaller at R  =  0.64 m, the heat exchange from the $T_{{\rm eh}}$ component  ∼0.5 keV heated by the electrostatic EBW with $\newcommand{\para}{\mathbin{\!/\mkern-5mu/\!}} N_{\para} \sim 2$ should be effective in increasing the density through increased ionization and sustaining the high-density plasma against radiation cooling.

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During the EC non-inductive plasma ramp-up, energetic electrons were created to generate the plasma current effectively. However, it has been rather difficult to heat bulk electrons in non-inductively ramped-up plasmas after creation of these energetic electrons [21]. The incident ECW power seems to couple into the highly energetic electrons easily even in the ray-scrambling situation with multiple wall reflections. Evaluation of the synergetic heating with higher harmonic resonances at  ∼500 keV tells us that heating on the highly energetic electrons is performed effectively with a feedback process in association with weaker heat exchanges or collisions between the energetic and bulk electrons. As $N_{\perp}$ becomes larger in the electrostatic wave, the $\newcommand{\e}{{\rm e}} \exp ( - \lambda_{{\rm h}})$ term could suppress the EBW absorption via the highly energetic electrons. Absorption was still effective enough in the higher-density plasma to heat the mildly energetic and bulk electrons through heat exchanges under stable equilibrium with highly energetic electrons. The effective EBW heating may start to increase the density and maintain an over-dense plasma even if the density is beyond the cutoff. Effective plasma production through the electrostatic EBW heating appeared as the SDJ of a bifurcation phenomenon.